# Modeling Wave Phenomena

This lab is designed to align with AAOT science outcome #1: Gather, comprehend, and communicate scientific and technical information in order to explore ideas, models, and solutions and generate further questions.

## MAterials

• writing utensil
• calculator
• digital device with spreadsheet program
• digital device with internet access

## Objectives

1. Analyze pulses to determine the speed of traveling waves on a real medium.
2. Analyze real travelling waves to determine amplitude, wavelength, frequency, wave number and full sinusoidal equation of  the wave.
3. Model the resonant frequencies of an open-ended system and predict the resonant mode of real standing wave.
4. Compare the actual resonant mode with that predicted by the model.
5. Predict the beat frequency produced by the superposition of two waves of different frequencies
6. Simulate the superposition of two waves of different frequencies to visualize beats.
7. Use the visualization to determine the beat frequency and compare to the predicted value.

## Experimental Methods

The following video shows a Vinycombe wave machine used to create travelling pulses, travelling waves, and standing waves. We will extract data from the video for further analysis. Watch the video before continuing with the lab.

## Analysis

### Pulse Analysis

1) The section of the video on positive pulses (near 1:16) shows the pulses travel from one side of the machine to the other in real time. Assume the wave machine is 2m long and that the video is being played at real speed.  Use a stopwatch to measure the time for a pulse to travel across the machine and then calculate the pulse speed. Reduce uncertainty by playing the video at 0.25 speed while you time, and then multiplying the time by 0.25. Record your values and show your work below. (The playback speed settings can be accessed by clicking the gear icon in the lower right of the player window).

### Travelling Wave Analysis

2) The section on travelling waves (near 3:25) shows a traveling wave. Time the oscillations of the hand used to generate the wave to determine the wave frequency. Again, reduce uncertainty by reducing playback speed. Record your values and show your work below.

3) Use the section when the traveling wave is frozen to calculate the wavelength. (You will need to use the known 2m length of the machine to find a conversion between actual distance in meters and distance measured on the screen you are using to watch the video). Record your values and show your work below.

4) From the frequency and wavelength data, calculate the expected wave speed. Show your work.

5) Do the pulse speed and the travelling wave speed appear to be the same? Calculate a percent difference.

6) Use your distance conversion factor to determine the amplitude of the wave. Show your work.

7) Write the complete traveling wave function for this wave, with numerical values for amplitude, angular frequency, and wavenumber. Show any work you to find these values.

### Standing Wave Analysis

8) The bars on the ends of the wave machine are free to oscillate and will form antinodes. The first resonant mode would have one node directly in the middle and antinodes directly on the ends. Draw a picture of such a standing a wave formed in the machine at a moment in time.

9) In terms of the machine length, L, what is the wavelength of the first resonant mode?

10) What is the first resonant frequency? Show your work. (Hint: You already know the wave speed.)

11) The second resonant mode will have two nodes, still with anti nodes at the ends. Draw a picture of such a standing a wave formed in the machine at a moment in time. What is the wavelength of this resonant mode?

12) What is the second resonant frequency? Show your work.

13) As you can see, this machine will support resonant modes that correspond to all harmonics of the first resonant frequency, just like a system that is fixed on both ends (as opposed to a system only fixed on one end that only supports odd harmonics). Write a function for the nth resonant frequencies of this machine in terms of the wave speed, machine length, and n.

14) Time the oscillations of the hand generating the waves that form standing waves (near 3:45). Determine the driving frequency. Record your values and show your work.

15) Is the driving frequency near one of the resonant frequencies? Which resonant frequency is nearest?

16) How many nodes should be present in the standing wave mode associated with the resonant frequency you found above, and what should the distance between them be? Explain/show your work. [Hint: Re-examine the pictures of the first few resonant modes to look for a pattern in number of nodes vs. resonant mode).

17) Use the video to determine the actual number of nodes and the distance between them. (You will need to use the known 2m length of the machine to find a conversion between actual distance in meters and distance measured on the screen you are using to watch the video). Record your values and show your work below.

18) How does the standing wave mode (number of nodes and distance between them) compare to the mode predicted from the resonant frequency (in 16)?

## Conclusions

19) Overall, does the free-ends model do well at describing the spatial behavior of this standing wave resonant modes?

## Further Questions

Standing waves are an example of a spatial interference pattern (the amplitude of oscillation changes across different locations in space, but the amplitude at those locations is constant in time). Beats are an example of a temporal interference pattern (the amplitude of oscillation at one location will change in time).

You can hear beats when you listen to two soundwaves of similar frequency (listening for and minimizing the beat frequency is how instruments are tuned). We will visualize the continuous arrival of two waves of different frequencies at a certain location in space. To do that, we simply need to calculate the displacement of the two waves at each point in time and apply the superposition principle to determine the resulting amplitude at each point in time.

20) Create a spreadsheet with time values in the first column running from 0 to 200 in increments of 1. In the second column calculate the displacement of a traveling wave at each time with period of 10 and amplitude of 5.  Choose the current location of the wave as x = 0 so that the oscillation at that point can be described by $Asin(\frac{2\pi}{T}t)$ for A = 5, T = 10.

21) Plot the displacement vs. time for this location. Choose a plot type that connects the points so you can visualize the signal. Be sure to title your graph and label the axes (we haven’t specified any units for time or displacement, so write “arb. units”, which is short for arbitrary units).

22) In the third column repeat the procedure for a wave with A = 5, T = 9. ($Asin(\frac{2\pi}{T}t)$ for A = 5, T = 10).

23) Calculate the frequencies of these two waves and their expected beat frequency? Show your work.

24) Make a fourth column that is the superposition of the first two waves to determine the resulting displacement at this location where the waves meet.

25) Make a whole new plot of the resulting displacement vs. time for this location. Choose a plot type that connects the points so you can visualize the resulting signal. Be sure to title your graph and label the axes (we haven’t specified any units for time or displacement, so write “arb. units”, which is short for arbitrary units).

26) Using the new graph, determine the period of the resulting wave envelope as the time from one peak to another and record below. This is the beat period you would hear if these were two sound waves.

27) Use the  beat period to calculate the beat frequency. Show your work.

28) How does the beat frequency compare to the expected beat frequency?